The Science of Powerball (Synopsis)

“I’ve done the calculation and your chances of winning the lottery are identical whether you play or not.” –Fran Lebowitz

Later today, the richest lottery drawing in history — the $1.5 billion Powerball jackpot — will take place. While many outlets are encouraging people to purchase as many tickets as possible, it’s important to run through the mathematics and find out what your expected value is for each ticket.

Image credit: E. Siegel, 2016.

Image credit: E. Siegel, 2016.

While a naive analysis shows that a jackpot in excess of about $245 million would lead to a break-even-or-better result, when you factor in taxes and split jackpots, you find that even for the $1.5 billion jackpot, your $2 ticket is only worth about $0.85.

Image credit: E. Siegel, 2016.

Image credit: E. Siegel, 2016.

It’s often been said that lotteries are taxes on the mathematically challenged; go find out why!

Comments

  1. #1 eric
    January 13, 2016

    Didn’t we already cover this the last time it got big? Obviously not this big though. This is quite an astounding historical event just because of the amount.

    While there’s no way to make it a good bet (i.e., above the breakeven point), you can make it a better bet by selecting numbers superstitious people are unlikely to select. That generally means numbers >30 since many people play birthdays or other dates, and numbers in sequence since laypeople tend to think sequences are less likely than scattered values. 46,47,48,49,50, 51 has the same crappy odds of winning as any other combo, but at least if you pick a ticket like that, you can console yourself with the idea that you likely won’t be sharing the winnings with many people if it hits. And don’t worry about sharing with other people who take my advice, the pool of “people who take eric’s advice” is quite small. Probably one. Somedays, not even one. 🙂

    Well, in any event good luck and best wishes to those who play.

  2. #2 dean
    United States
    January 13, 2016

    I’ve always wondered about the appropriateness of the “TAX on the mathematically challenged” line. A lottery ticket purchase is a choice people make: they can buy tickets or not as they wish. That isn’t the case with a tax.

  3. #3 Wow
    January 13, 2016

    You can choose with taxes too. Get a crappy low paid job and you won’t have to pay tax. Or move to somewhere where there is no tax.

    However, unlike the lottery, where you get nothing if you don’t win, you get something with your taxes every time you have paid, and even when you haven’t.

    Paying taxes is no different from paying the butcher or baker for their produce, or paying the landlord the rent.

  4. #4 dean
    United States
    January 13, 2016

    My comment wasn’t a statement against taxes – it was a statement about the comparison of lottery cost with taxes.
    Having a crappy low paying job doesn’t keep you from paying taxes – you still pay taxes on the items you purchase.

  5. #5 Wow
    January 13, 2016

    Yeah, but the comparison is wrong for reasons other than the ones you cried off on.

    You get taken for a ride with the lottery, getting nothing back.
    You get something back for your taxes.

    But in both cases you can decide to not pay.

    You claimed you could decide to pay with the lottery but not taxes. This is not the case at all. You have two methods at least, and one of them allows you to get stuff FREE.

    “you still pay taxes on the items you purchase.”

    So don’t purchase anything. This is still a choice.

  6. #6 dean
    January 13, 2016

    “So don’t purchase anything. This is still a choice.”

    No, purchasing food (for example) is not a choice. The fact that there are other ways to examine the difference between purchasing tickets for a lottery and paying taxes than the one I mentioned might be true, but it doesn’t change my comment.

  7. #7 Wow
    January 13, 2016

    “No, purchasing food (for example) is not a choice”

    (Ignoring that the response to this is “Yes it is.”)

    All you’re REALLY saying here is that buying food is like paying taxes in your opinion: you don’t get a choice of whether you do pay.

    Don’t hear “food” as being related to taxes anywhere.

    “The fact that there are other ways to examine the difference between purchasing tickets for a lottery and paying taxes than the one I mentioned might be true,”

    It is.

    ” but it doesn’t change my comment.”

    Nonsequitur.

    Some other differences being true doesn’t make your comment right either.

  8. #8 dean
    January 13, 2016

    Nothing you’ve said explains why the choice of paying for a lottery ticket should be viewed in any way the same as paying taxes. The notion that purchases of food (or other necessities I didn’t mention) are choices doesn’t hold water, nor does your implication that there is some subclass of society that pays no taxes at all.

  9. #9 Wow
    January 13, 2016

    “Nothing you’ve said explains why the choice of paying for a lottery ticket should be viewed in any way the same as paying taxes.”

    Nothing about not being looked at in the same way makes the difference that you can choose to pay the lottery ticket but can’t choose to pay taxes.

    “The notion that purchases of food (or other necessities I didn’t mention) are choices doesn’t hold water”

    Yes it does.

    But even if it weren’t valid, your problem is now that you think of taxes being like buying food.

    “nor does your implication that there is some subclass of society that pays no taxes at all.”

    I’ve not said that there is. Though I can posit that many don’t. However, become a subsistence farmer in Somalia. No taxes and you grow your own food.

    There’s nothing stopping you doing that except preferring to be a US citizen paying US taxes (and doing the job you are employed in that doesn’t supply you with all your basic needs) and not having to live in Somalia as a subsistence farmer.

    You CHOOSE to “have to” pay those taxes.

    Just like you choose to pay the lottery.

  10. #10 Wow
    January 13, 2016

    “Nothing you’ve said explains why the choice of paying for a lottery ticket should be viewed in any way the same as paying taxes.”

    And you’re really not listening. I’ve not said they’re the same. I’ve said they’re DIFFERENT. The difference is that if you pay the lottery, you probably won’t get anything at all for it. If you pay your tax, you DO *definitely* get something for it.

    You don’t get to “choose” to pay taxes because you can’t choose not to use the roads, the protection of the building codes protecting your home from being dangerous, from the EPA making sure that the area you live in isn’t polluted, the police and justice system seeing that there’s some limit on criminality, the army from ensuring that no mexican invasion takes over your country, regulations that ensure the banks can’t just take your money or your boss fails to pay you.

    You can’t choose NOT to have those, so why should you choose not to pay for them?

    If you’re susbsistence and in the middle of nowhere, therefore not using the roads, clean water, power or courts system, then you won’t be paying taxes.

    You can choose to live with other humans and not slave away to get the bare minimum of food and have to make everything you need yourself and fix everything that breaks, or you can choose not to pay tax. And you choose the former.

    So don’t go crying how you can’t choose: you’re lying.

  11. #11 Wow
    January 13, 2016

    If you’re worried about choosing to pay taxes on necessities, then get the law changed so that sales taxes (or just those on necessities) disappear and that the various income taxes (or estate taxes) are increased to make the shortfall up.

    The UK has a zero rate VAT on food and similar. Only luxury classed foods are taxed with VAT. And, yes, we do have slightly higher income taxes, but you choose what level of tax you want when you go for a particular wage. You don’t have to be “forced” to pay “tax” because you’ll starve to death otherwise.

  12. #12 Carl
    USA
    January 13, 2016

    My numbers have been 1, 2, 3, 4, 5, and 6 for every lottery on the planet, for over 20 years now (shrink sequence to fit the # of balls of varying lotteries). I don’t buy a ticket, never have.

    My approach is that I can ignore them – if my winning numbers ever hit, it will be front page news and riots are likely. Then I’ll say “Dang it! If only I’d bought a ticket!”. But until that time, I can multiply the # of weeks by the cost / week to play, and rightfully claim to be in the black by that amount.

    In my home state lottery, I’m up by over $3,000!

  13. #13 Carl
    USA
    January 13, 2016

    Oh, and dean – you are completely right. The lottery is not a tax.

    wow must be having fun trolling you, since he often tells posters to Google for available information before bothering people here. Taxes are compulsory. Google it, wow.

  14. #14 Narad
    January 13, 2016

    I’m coincidentally reminded of the recent “calculation” that virological gain-of-function research carries a risk of 54 deaths per lab-year.* What this figure conceals (among other things) is a roughly 55,000 yr timescale.

    * Not to get too far into the minutiae, but this doesn’t seem to actually appear in the RBA, but rather to follow from the public comments of Lynn C. Foltz, who also included the odd statement that “the 0.4% value [of a single laboratory-acquired infection seeding a pandemic] is likely 1/0.02 = 50-times higher due to eliminating this intermediate local-outbreak step.” So it’s 200%?

    The RBA is here, for anyone whose appetite for probabilities has been whetted by Poweball:

    h[]tp://osp.od.nih.gov/sites/default/files/Risk%20and%20Benefit%20Analysis%20of%20Gain%20of%20Function%20Research%20-%20Draft%20Final%20Report.pdf

    The public comments are here (Klotz and Lipsitch are the standouts):

    h[]tp://osp.od.nih.gov/sites/default/files/Compiled%20Written%20Public%20Comments.pdf

  15. #15 Narad
    January 13, 2016

    ^ Mostly chapter 6 of the RBA, and for mammalian-transmissible highly pathogenic avian influenza, BTW.

  16. #16 Narad
    January 13, 2016

    I’ve always wondered about the appropriateness of the “TAX on the mathematically challenged” line.

    *shrug*

    It’s a metaphor (although I hear it more often referred to as a “stupidity tax”). The basic point is that in the real world, lotteries tend to be a regressive way of pulling money into state coffers. If you want something that’s more directly comparable, look at sin taxes.

  17. #17 Carl
    USA
    January 13, 2016

    IMO, a lottery ticket is entertainment. For $3, you can fantasize about some uber-wealthy future state. Similar to paying to see a movie, except this one is in your imagination.

    From that perspective, it’s not regressive. It’s a cheap form of entertainment that appeals to poor people more than it does to well-off people. It isn’t sinister, except in those cases where people suffer from a gambling addiction.

  18. #18 dean
    United States
    January 13, 2016

    “It’s a metaphor”
    I understand that – my comment was simply that I don’t think it’s a good one. Sheesh.

    I just covered the probability and expected value calculations in my intro class on Monday. Yesterday one student gave me an article that talked about the “coverage” of the different possible plays: as Ethan says, there are 292,000,000 possible ticket plays. Last Saturday night, according to lottery officials, 75% of those possibilities were played. The forecast is that tonight approximately 80% will be played. That is a metric butt-load of tickets.

  19. #19 Narad
    January 13, 2016

    “Lottery expert” Dawn Nettles is hilariously clueless, BTW.

    “Everyone should choose their own numbers…. If players had created their own Quick Picks on Saturday night, I think all combinations would have been sold and we would have had a winner.”

  20. #20 Narad
    January 13, 2016

    I understand that – my comment was simply that I don’t think it’s a good one. Sheesh.

    OK, OK, I hadn’t had any coffee yet. Sheesh, yourself.

    @Carl:

    IMO, a lottery ticket is entertainment. For $3, you can fantasize about some uber-wealthy future state. Similar to paying to see a movie, except this one is in your imagination.

    Ideally, sure. I’ve bought Powerball tickets about twice in my life, and I can also find hours of entertainment in devising screwball numerological schemes for making the picks. This time around I also got to review my counting statistics and have some reading material about the relevant tax law* (e.g., here; “[k]eep in mind how unused to complexity most lottery winners are,” which seems to be a bit haughty). But…

    From that perspective, it’s not regressive. It’s a cheap form of entertainment that appeals to poor people more than it does to well-off people. It isn’t sinister, except in those cases where people suffer from a gambling addiction.

    I don’t think that either prong really holds, based on the anecdata of my having lived in a diverse urban neighborhood for a long time (oh, and having repeatedly failed to quit smoking). The people that I see buying lottery tickets are generally playing the lower stakes games such as pick 3 (with six bet types) and scratch-off cards.

    There’s no need to invoke “gambling addiction” to explain routinized behavior, and once it’s routinized, the “them po’ folks get a kick out of it” prong falls as well. Powerball is a multistate extravaganza, but I’m willing to venture – without actually looking at the books – that my state’s lottery lives and breathes on the “penny ante” games.

    This is where the social Darwinism unwraps itself from the metaphors, which is where I was trying to go in the first place. I’d never heard “tax on the mathematically challenged” before, but it seems all the more obnoxious for its cutesiness.

  21. #21 Narad
    January 13, 2016

    ^ Back to Powerball’s entertainment value, though, I left off that the few times I’ve played, there was a pretty consistent payout in terms of chatting with pleasantly interesting characters while waiting in line. YMMV.

  22. #22 eric
    January 13, 2016

    “Everyone should choose their own numbers…. If players had created their own Quick Picks on Saturday night, I think all combinations would have been sold and we would have had a winner.”

    I do wonder whether the quick pick machines are truly random or just pseudo random. I think if they were truly random, we would’ve heard some news story about someone suing because the machine gave them 1,2,3,4,5,6. I bet that just as weather.com makes their forecasts less accurate for public acceptance purposes, the quick pick machines are somewhat less than random for public acceptance purposes.

  23. #23 Wow
    January 14, 2016

    “Oh, and dean – you are completely right. The lottery is not a tax. ”

    So I’m right too. I say its not a tax:

    The difference is that if you pay the lottery, you probably won’t get anything at all for it. If you pay your tax, you DO *definitely* get something for it.

    “wow must be having fun trolling you”

    By agreeing they’re not the same???? Obviously you must be trolling dean…

    Or an asshat.

  24. #24 Wow
    January 14, 2016

    “the few times I’ve played, there was a pretty consistent payout in terms of chatting with pleasantly interesting characters while waiting in line”

    You can do that with any shopping visit. Or just walking in the street.

  25. #25 Wow
    January 14, 2016

    “Taxes are compulsory. Google it, wow.”

    Taxes are not compulsory. Google it,Carl. Or read the evidence I have given above.

  26. #26 Carl
    USA
    January 14, 2016

    The various “tax” metaphors are actually helpful, where they remind people that their lottery ticket money is probably wasted.

    “If it only helps one person make a better fiscal decision, it’s worth it.” Isn’t that how the metaphrase goes?

  27. #27 Carl
    USA
    January 14, 2016

    Wow, in this case you are definitely trolling. dean @2 wondered at the “tax” angle of the lottery metaphor, and you argued @3 that taxes were optional.

    You spend the rest of your posts fighting to prove you’re right, digging a deeper hole as you go. By @7 you are needlessly rude, @10 you start name calling (i.e. dean is a liar). None of that behavior was provoked.

    The dictionary definition of “tax” is clear; they are compulsory. Here are the ones that I reference (search phrase “definition of tax”):

    Google definition
    dictionary.reference.com/browse/tax
    dictionary.reference.com/browse/taxes
    http://www.merriam-webster.com/dictionary/tax
    http://www.investopedia.com/terms/t/taxes.asp

    Those are the top 5 search returns, and I’ve only stopped posting more because their is no point – they all agree.

    My analysis – you like to argue and call names. You like to “win” arguments, and try to provoke them. You think you are smart enough to twist logic and language to “win” arguments, even when your position is wrong. Other than the name calling, you like to play “debate team.” The name calling thing is just you.

    You seem to be smart on many science topics that come up here. You’ve helped answer questions and educated posters, though you are often abrasive in the process (e.g. “Google it”). In this case you are just plain wrong.

  28. #28 dean
    January 14, 2016

    I do wonder whether the quick pick machines are truly random or just pseudo random. I think if they were truly random, we would’ve heard some news story about someone suing because the machine gave them 1,2,3,4,5,6. I bet that just as weather.com makes their forecasts less accurate for public acceptance purposes, the quick pick machines are somewhat less than random for public acceptance purposes.

    They are psuedo-random, since they are generated by a mechanical process. That said, the occurrences of the numbers pass tests for randomness, so they are as “random” as we can make them.

  29. #29 eric
    January 14, 2016

    That said, the occurrences of the numbers pass tests for randomness, so they are as “random” as we can make them.

    Well then, I’m surprised nobody’s sued about getting a sequence. Maybe I’m cynical about numeracy, but I expect a lot of ticket buyers would interpret a 1,2,3,4,5,6 ticket as “machine broken, I want my money back.”

  30. #30 Carl
    USA
    January 14, 2016

    Hey! That 1,2,3,4,5,6 ticket is mine! (see #12)

  31. #31 Chris Mannering
    January 16, 2016

    That line about playing the lottery indicating stupidity, comes out easily the stupidest stupid of them all. People don’t buy a lottery ticket for the odds. They do it for inexpensive fun. Or hope. For home entertainment of the children. For facilitating god’s miracle gift their way, just in case he wants to do that, only there’s never any good openings.

    The latter reason has really caught on in a big way since hundreds of worshippers heard God’s booming words from the mouth of an evangelical speaking in tongues. Like totally for real and someone caught the audio on their Galaxy S5, sending it totally crazily viral with 40 million views on the same day. A spinster had got wind of the stupidest stupid clever turnip 8th grade probabiliificationing and took true umbrage in the direction of it and then took it out on God who had not miracle her a win eve, after playing with high fidelity since she was 37. She was screaming and crying out and banging her head against the vicars pocketed St James edition Holy New Testimant. How could He have forsaken her and now see her sodden in clever clog statistical peroratives She threw her arms up with her best profile side pumping a plum silhohette in afore mentioned Galaxy S5, which right at that point did God boom out “you never did buy a ticket you stingy old spinster”
    You’ve been shaggy dogged baby

  32. #32 Wow
    January 17, 2016

    “That line about playing the lottery indicating stupidity, comes out easily the stupidest stupid of them all”

    No it doesn’t.

    Does spending $3 to expect ~50c back sound like a sensible thing to do?

  33. #33 Vincenzo Romano
    January 20, 2016

    The best provable way not to loose at the lotteries is NOT TO BUY TICKEYS. And you can even win!

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